The quantification of motion not only occurred in medieval dynamics, but also in kinematics, which relates motion to time and space traversed. This type ofapproach should be seen in the light of the introduction of logic and geometry in natural philosophy.
Scholars at Paris had a predilection for providing semantic analyses of the terms in which their physical problems were formulated. Terms such as ‘‘motion,’’ ‘‘nature,’’ ‘‘change,’’ “alteration,” ‘‘point,’’ ‘‘space,’’ ‘‘time,’’ and ‘‘instant,’’ for example, were submitted to an analysis that employed all the logical techniques available. Especially the theory of supposition was fundamental. It was a tool that analyzed a term’s reference within the context of a proposition and in this way determined the meaning and truth of that proposition. The supposition theory provided, for instance, different semantic analyses of the propositions ‘‘man is a species’’ and ‘‘man is a three-letter word.’’ Another much-used semantic tool was the analysis of a term’s position within a proposition, the ‘‘word-order,’’ so to speak. This aspect was expressed in the technical vocabulary of distinguishing between the categorematic and the syncategorematic use of a term. Since these semantic aspects significantly affected the truth-value of the propositions in which the physical problems were stated, they had a bearing on the solution of these problems (see the entries on Supposition Theory; Syncategoremata and Terms, Properties of in this volume).
At Oxford, the application of logic in kinematics was blended with mathematical techniques. As a matter of fact, one of the distinctive features of most of the Calculators’ treatises is that they originated out of a logical, disputational context. This is especially true for the Sophismata, collections of counterintuitive statements called ‘‘sophisms’’ that served as examples to illustrate semantic theories. Often, sophismata have a purely logical character, but especially at Oxford a new genre originated, that of the mathematical-physical sophisms. This emphasis on logico-mathematical techniques, rather than directly on physical theory, is present in the treatises of Heytesbury and Swineshead and also in the Sophismata of Richard Kilvington. A typical example of the type of problems discussed there is the truth of sophism 34, ‘‘Plato can move uniformly during some time and as fast as Socrates now moves’’ (see the entries on Sophisms and Richard Kilvington in this volume).
The quantification of kinematic aspects of motion was introduced in the context of the so-called doctrine of the latitude of forms (latitudo formarum), a theory that was developed to deal with the different degrees that may be assigned to one and the same quality. For instance, one banana can be more yellow than another one, or one object hotter than another one. The idea that qualities in a subject can exist in varying degrees was first expressed in Aristotle’s Categories (10 b 26 and following). It gave rise to two problem areas that were only very loosely connected. The first one, called the problem of the intension and remission of forms in medieval terminology, developed further the idea that qualities can become more or less, that is, that they can undergo intensification and remission (intensio et remissio). It focused on ontological issues such as the search for the subject of the strengthening or weakening: was it the quality itself, or the subject’s participation in the quality? And how was a quality intensified or weakened? The two most prominent alternatives that had emerged by the fourteenth century were the succession theory and the addition theory.
Discussions of the intension and remission of forms tacitly assumed that a quality was uniformly distributed in a subject and uniformly changed over time. However, precisely this assumption was further investigated in the second problem area, namely that of the latitude of forms (latitudo formarum). This problem area studied the phenomenon of qualitative changes in a subject from the perspective of space and time. The point of departure was the idea that qualities can be non-uniformly, or difformly (difformiter) according to medieval terminology, distributed over a given subject and that they can change difformly. The scholars at Oxford focused on the problem of measuring these qualities, especially the uniformly difform qualities, that is, that class of qualities that strengthened or weakened at a constant rate from one end of the subject to the other. By analogy, they addressed questions of measures of other types of change, especially of local motion. It was in this context of measuring qualities and motions that the concept of latitude came to play a pivotal role. It is important to note that these discussions often were hypothetical (see the entry on Intension and Remission of Forms in this volume).
The philosophical analysis of the degrees of qualities helped to develop the idea of velocity as a magnitude to which can be attributed a numerical value and by which motions can be measured. The idea had been quite alien to Aristotle, who had conceived of velocity as an unquantifiable concept. One theorem that developed out of the latitude of forms context has been characterized by at least one historian of science as ‘‘probably the most
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Outstanding single medieval contribution to the history of mathematical physics’’: the mean speed theorem (Grant 1996:101). This theorem measures uniformly accelerated motions with respect to the spaces traversed by comparing them with uniform motions. The mean speed theorem states that a mobile moving with a uniformly accelerated motion covers the same space in a given time as it would if it moved for the same time with a uniform speed equal to the speed at the middle instant of the duration of its acceleration. The mean speed theorem has attracted much scholarly attention, because it was mentioned by Galilei in his Discourses on Two New Sciences (Day 3, theorem 1) and applied to the free fall of bodies, which is an example in nature of a uniformly accelerated (i. e., uniformly difform) motion.
The Calculators provided many proofs of the theorem, but none was as easy to visualize as Oresme’s geometrical proof, which was also employed by Galilei. Oresme noted that a right triangle is equal in area to a rectangle whose height is the mean height of the triangle. In this way he graphically compared the quantity of a uniformly difform quality to that of a uniform quality of mean intensity. Other Parisians besides Oresme were also well acquainted with the Oxford’s Calculators’ discussions of the measure of the effects of motion, as is clear from their discussions in commentaries on the Physics. As Edith Sylla has convincingly argued, the aim of the logical and geometrical discussion of kinematics was to instruct students to think and argue clearly and exactly. Newton’s Principia mathematica philosophiae naturalis still carries the echos of medieval natural philosophy in its title, although the emphasis had now definitely shifted toward the mathematical approach.
See also: > Adelard of Bath > Albert of Saxony > Arabic Texts: Natural Philosophy, Latin Translations of > Aristo-telianism in the Greek, Latin, Syriac, Arabic, and Hebrew Traditions > Aristotle, Arabic > Atomism > De caelo, Commentaries on Aristotle’s > De generatione et corruptione, Commentaries on Aristotle’s > al-FarabI, Abu Nasr > al-FarabI, Latin Translations of > Form and Matter > Francis of Marchia > Ibn Rushd (Averroes), Latin Translations of > Ibn Rushd, Muhammad ibn Ahmad al-Hafld (Averroes) > Ibn Sina, Abu 'All (Avicenna) > Ibn Sina (Avicenna), Latin Translations of
> Intension and Remission of Forms > John Buridan
> John Dumbleton > John Duns Scotus > Marsilius of Inghen > Nicholas Oresme > Oxford Calculators > Parva naturalia, Commentaries on Aristotle’s > Philosophical Psychology > Posterior Analytics, Commentaries on Aristotle’s > Richard Kilvington > Richard Swineshead
> Roger Bacon > Sophisms > Supposition Theory
> Syncategoremata > Terms, Properties of > Thomas Bradwardine > Translations from Greek into Arabic
> Universities and Philosophy > William Heytesbury
World History
